Lucidi Duino 11 - ELETTRA

Paolo Fornasini Dipartimento di Fisica Università di Trento

Basic attenuation mechanisms for X-rays Paolo Fornasini

Univ. Trento

outgoing beam

Elastic scattering

Inelastic scattering

Photo-electrons

Auger electrons

Fluorescence Absorption

Scattering

Beam attenuation Incoming beam

X-RAYS and X-ray techniques Paolo Fornasini

Univ. Trento

X-rays

10-12 10-10 10-8 10-6 10-4 10-2 1 Å 1 µm

λ (m)

U.V. I.R.

E[keV] = 12.4λ[Å]

Scattering

Bragg angle

λ = 2dsinθ

Spectroscopy

nEnergy

Imaging

Structural techniques

Scattering

Paolo Fornasini

Univ. Trento

X-rays

Neutrons

Spectroscopy

Energy

Long-range order Short-range order

•  X-rays absorption – phenomenology •  X-rays absorption - theory •  EXAFS: theoretical background •  EXAFS experiments •  EXAFS: data analysis, examples

X-rays absorption - phenomenology

X-rays, 1897

Attenuation of X-rays Paolo Fornasini

Univ. Trento

source monochromator

sample

detectors

Φ0 Φ

Φ = Φ0 exp −µ ω( ) x[ ]

µ ω( ) =1xlnΦ0

Exponential attenuation

Attenuation coefficient

Atomic cross sections Paolo Fornasini

Univ. Trento

10-2 100 102 104 106

E (keV)

Photoelectric absorption

Coherent sc.

Incoherent sc.

Pair prod.nucl. field

elec. field

32 - Ge

µ ω( ) =NaρA

µa ω( )

Photo-electric absorption Paolo Fornasini

Univ. Trento

1 10 100

µ/ρ (cm2/g)

60-Nd, 65-Tb, 70-Yb

hω (keV)

1 10 100

µ/ρ (cm2/g)

Photon energy (keV)

32 - Ge

τ ω( )∝ Z 4

ω( )3

Excitation and ionization

-10000

Ep (eV)

- 13,6

0 Ep (eV)

1- Hydrogen 29 - Copper

Paolo Fornasini

Univ. Trento

unoccupied free-levels

unoccupied bound-levels

Absorption edges

1 10 100

µ/ρ (cm2/g)

60-Nd, 65-Tb, 70-Yb

hω (keV)

Z > 9 Z > 29

L2 2p1/2 L3 2p3/2

M1 3s M3 3p3/2

M2 3p1/2 M4 3d3/2 M5 3d5/2

Paolo Fornasini

Univ. Trento

0 20 40 60 80 100

Physical Review 16, 202 (1920)

Absorption edge fine structure

Atomic gases: edge fine structure

-6 -4 -2 0 2

E - Eb (eV)

Ar Eb = 3205.9 eV

Paolo Fornasini

Univ. Trento

Core level Unoccupied bound levels (Rydberg levels)

L. G. Parrat, Phys. Rev. 56, 295 (1939)

Atomic gases: smooth absorption coefficient

14.2 14.5 14.8 15.1Photon energy (keV)

Paolo Fornasini

Univ. Trento

Core level Unoccupied free levels (continuum)

Edge Extended region

Molecular gases: Fine structure Paolo

Fornasini Univ. Trento

K-shell - gas-phase N2

400 405 410 415 420 Photon energy (eV)

Core level Unoccupied bound levels Unoccupied free levels

C.T. Chen and F. Sette, Phys. Rev. A 40 (1989)

Vibrational levels

Rydberg series

Double excit.

Shape reson.

Condensed systems: Fine structure

11 11.5 12Photon energy (keV)

Paolo Fornasini

Univ. Trento

Core level Unoccupied bound levels Unoccupied free levels

Extended region

XAFS: X-ray Absorption Fine Structure

X-ray Absorption Near Edge StructureNear Edge X-ray Absorption Fine Structure

Paolo Fornasini

Univ. Trento

XANESNEXAFS

Extended X-ray AbsorptionFine Structure

XAFS: edge and pre-edge

5 4.98 4.96 5.02 5.04 Photon energy (keV)

Edge Pre-edge

Lectures on XAFS

EXAFS •  introduction •  basic theory •  experiments •  data analysis

P. Fornasini

XANES phenomenological approach

C. Meneghini

XAFS multiple scattering approach

M. Benfatto

Applications

XAFS & Materials science F. Boscherini

SR & Earth science S. Quartieri

SR & Cultural heritage S. Quartieri

SR & Chemistry A. Martorana

SR & Environmental science P. Lattanzi

X-rays absorption - theory

X-ray attenuation

• Linear attenuation coefficient

• Mass attenuation coefficient

µρ=Na

Elements:

= xµρ

+ yµρ

Ai = atomic weights, M = molecular weight

Chemical compounds PxQy...

Φ0 Φ

Energy density u = ε0E02

2=ε0ω

µ ω( ) = −1ududx

=1xlnΦ0

Φ=NaρA

µ a ω( )

Na = Avogadro numberA = atomic weightr = mass density

µa = atomic cross section

Paolo Fornasini

Univ. Trento

Absorption coefficient

Pif = ?

Paolo Fornasini

Univ. Trento

Absorption coefficient

Atomic density

Vector potential amplitude

Transition probability per unit time

i = initial ground state f = final excited states

E f − Ei = ω

µ ω( ) =2

ε0ωA02 n

dPifdtf

Radiation-matter interaction

Pif = ?

Initial atomic state

Stationary ground state Ψi

Final atomic state

Stationary excited state Ψf

Interaction

Ψ t( )

Paolo Fornasini

Univ. Trento

Perturbation approach

Paolo Fornasini

Univ. Trento

System = atom (quantum treatment)

Weak perturbation = electromagnetic field (classical treatment)

Hamiltonian for atom in e.m. field Paolo Fornasini

Univ. Trento

∇ ⋅ A =0

Φ = 0 J = 0

Radiation gauge

E = ∂

A /∂t

B = ∇ × A

∇ ⋅ A = A ⋅ ∇

Vector potential

H =12m

p j − q A r j ,t( )[ ]

qm s j ⋅ B r j ,t( )

∑ + V r 1… r N( )

2m+ V r 1…

r N( )

j∑ +

em p j ⋅ A r j ,t( ) +

em s j ⋅ B r j ,t( ) +

2mA2 r j ,t( )

Unperturbed Interaction

e-e and e-p interact.

Sum over electrons Electron

q = −e < 0

Interaction Hamiltonian

Sum over electrons

A = Re A0 ˆ η ei(

k ⋅ r −ωt )[ ]

Paolo Fornasini

Univ. Trento

HI =em p j ⋅ A r j ,t( ) +

em s j ⋅ B r j ,t( ) +

2mA2 r j ,t( )

Relevant terms

Time dependence:

sinusoidal perturbation frequency ω

Transition to continuum states Paolo Fornasini

Univ. Trento Time dependent perturbation theory

Paolo Fornasini

Univ. Trento

Density of states

dPifdE f

= ρ(E f ) Ψf HI Ψi

2t δ E f − Ei − ω( ) 2π

Probability density

Time Energy conservation

1st-order perturbation

Transition to continuum states

''Golden rule''

HI =em p j ⋅ A rj ,t( )

wif =πe2

m2 Ao2

Ψi ei k ⋅ r j ˆ η ⋅ ∇ jj∑ Ψf

2ρ E f( ) δ E f − Ei − ω( )

A = Re A0 ˆ η ei

k ⋅ r [ ]

Paolo Fornasini

Univ. Trento

wif =dPifdE f dt

= ρ(E f ) Ψf HI Ψi

2δ E f − Ei − ω( ) 2π

Probability density per unit time

Golden rule

Initial atomic state

Stationary ground state Ψi

Final atomic state

Stationary excited state ΨfInteraction

with e.m. field

Time-dependent perturbation theory (1st-order)

matrix element density of final states

wif ∝ Ψi HI Ψf

2ρ E f( )

Paolo Fornasini

Univ. Trento

E f − Ei = ω

One-electron approximation

µtot ω( ) = µ el ω( ) + µinel ω( )

•  1 core electron excited •  N-1 passive electrons relaxed

•  1 core electron excited •  Other electrons excited

EXAFS coherent signal

µel ω( )∝ ΨiN−1ψ i ei

k ⋅ r ˆ η ⋅ p ψ fΨf

N−12ρ ε f( )

Paolo Fornasini

Univ. Trento

Electric dipole approximation

µel ω( ) ∝ ΨiN−1ψ i ˆ η ⋅

r ψ fΨf

N−12

ei k ⋅ r = 1+ i

k ⋅ r −… ≈ 1

HI ∝ ei k ⋅ r ˆ η ⋅ p ≈ ˆ η ⋅

p = ω 2 ˆ η ⋅ r

Δ = ± 1 Δs = 0Δj = 0,±1, Δm = 0,± 1

Dipole selection rules:

Paolo Fornasini

Univ. Trento

Validity of dipole approximation Paolo Fornasini

Univ. Trento

ei k ⋅ r = 1+ i

k ⋅ r −… ≈ 1

λ >> 2πr ⇔ kr <<1

10 30 50 70

K edgekr

r ≈ a0Z

a0 = 0.53Å( )

λ ≈1Z 2

1s orbital - K edge

Beyond the electric dipole approximation Paolo

HI =em p j ⋅ A r j ,t( ) +

em s j ⋅ B r j ,t( ) +

2mA2 r j ,t( )

For one electron:

Electric dipole

Electric quadrupole Magnetic dipole

Spin magnetic moment

Orbital magnetic moment

2-photon processes

ei k ⋅ r = 1+ i

k ⋅ r −…

µel ω( )∝ ψi ˆ η ⋅ r ψ f

2ρ ε f( ) Ψi

N−1 ΨfN−1 2

Sudden approximation

ΨN −1ψ = Ψ N −1 ψ

1 active electron N-1 passive electrons

S02 ≈ 0.6 ÷ 0.9

Structuralinformation

No interaction between photoelectron and passive electrons

Paolo Fornasini

Univ. Trento

The final state

Molecular orbitals theories

Band theories

Multiple scattering approach

Single scattering approximation

Fine structure

Basic EXAFS mechanism

at the core site

EXAFS: the mechanism

Superpositionat the core site.

Photo-electron propagation.

Photo-electron back-scattering.

X-ray photon absorption.Photo-electron emission.

Modulation ofabsorption coefficient -0.4

0 5 10 15 20

k χ(k)

Photo-electron wavenumber (Å-1)

Paolo Fornasini

Univ. Trento

De-excitation mechanisms

Radiative: fluorescence

Non-radiative: Auger

X e- e-

0 20 40 60 80 100

Fluorescence yield

Paolo Fornasini

Univ. Trento

Core-hole lifetime

20 40 60 80

Γh (eV)

K edges

L3 edges

Lifetime of the excited stateτh~10-16 -10-15 s

Energy width of the excited state

τh ≈1/Γh

Contribution tophoto-electron life-time

Energy resolutionof XAFS spectra

Paolo Fornasini

Univ. Trento

EXAFS: theoretical background

2m2 hν − Eb( ) =

2m2 ε = k =

Photon → photo-electron

11 11.5 12hν (keV)

Photon energy Photo-electron wavenumber

0 5 10 15k (Å-1)

χ (k)

Paolo Fornasini

Univ. Trento

Angular emission of photo-electron Paolo Fornasini

Univ. Trento

Photoelectron emission

Photon polarisation

asimmetry parameter

N(θ)∝1+β23cos2θ −1( )

N(θ) ∝ 3cos2θ = 3 ˆ η ⋅ ˆ r 2€

β = 2Emission from s orbitals

Photo-electron parameters

0 500 1000 1500

εf (eV)0

0 500 1000 1500

λ (Å

εf (eV)

Wave-number Wave-length

Energy

Paolo Fornasini

Univ. Trento

EXAFS normalisation

µ ω( )∝ ψ i ˆ η ⋅ r ψ f

χ(k) =µ − µ 0

µ0 ω( )∝ ψ i ˆ η ⋅ r ψ f

Paolo Fornasini

Univ. Trento

The EXAFS function (a)

0 4 8 12 16k (Å-1)

χ (k)

µ0 ω( )∝ ψ i ˆ η ⋅ r ψ f

µ ω( )∝ ψ i ˆ η ⋅ r ψ f

ψ f = ψ f0 + δψ f

(weak interaction) €

χ(k) =µ − µ 0

Paolo Fornasini

Univ. Trento

The EXAFS function (b)

0 4 8 12 16k (Å-1)

χ (k)

χ(k) =µ − µ 0

Quantum states → wavefunctions

χ k( ) =2Re d r ψi ˆ η ⋅

r ψ f0*( )∫ ψi

* ˆ η ⋅ r δψ f( )

d r ψi* ˆ η ⋅ r ψ f

Core orbital = source & detector

Paolo Fornasini

Univ. Trento

EXAFS: Two-atomic system (a)

•  Scattering theory in plane-wave approximation

•  Muffin tin potential

Paolo Fornasini

Univ. Trento

II I III

Absorber Scatterer

δψ f ∝ψ f0 i eiδ

exp ikR( )2kR

f k,π( )exp ikR( )

At the absorber core site

Propagators

Interaction

δψ f ∝ψ f0 i eiδ

exp ikR( )2kR

f k,π( )exp ikR( )

EXAFS: Two-atomic system (b)

χ k( ) = 3 ˆ η ⋅ ˆ R 2 1

kR2 Im f k,π( ) exp 2iδ1( ) exp 2ikR( )[ ]

spherical wave attenuation

back & forth path

central atom phase-shift

back-scattering amplitude

polarisation

Paolo Fornasini

Univ. Trento

χ k( ) =2Re d r ψi ˆ η ⋅

r ψ f0*( )∫ ψi

* ˆ η ⋅ r δψ f( )

d r ψi* ˆ η ⋅ r ψ f

Basic interference effect

χ k( ) = 3 ˆ η ⋅ R 1

kR2 Im f k,π( ) exp 2iδ1( ) exp 2ikR( )[ ]

χ k( ) = 3 ˆ η ⋅ ˆ R 2 1

kR2 f k,π( ) sin 2kR + φ k( )[ ]€

f k,π( ) e2iδ = f k,π( ) eiφ

0 5 10 15 20k (Å-1)

R = 2 Å

R = 4 Å

EXAFS frequency

Inter-atomic distance

Paolo Fornasini

Univ. Trento

Amplitudes and phase-shifts

[Calculated by FEFF 6.01] Z dependence

Central-atom Back-scattering

Amplitude Phase-shift Phase-shift

Paolo Fornasini

Univ. Trento

0 5 10 15 20

Back-scattering amplitude(a.u.)

k (Å-1)

32 - Ge

78 - Pt

0 5 10 15 20k (Å-1)

32 - Ge

78 - Pt

Back-scattering phase-shift(rad)

0 5 10 15 20k

32 - Ge

78 - Pt

Central atom phase-shift

k (Å-1)

Many-atomic systems Paolo Fornasini

Univ. Trento

Radial Distribution Functions

Coordination shells

Crystals Amorphous systems

Coordination shells

Coordination number

χ k( ) =1k

Nsshell∑ Im fs k,π( ) e2iδ1 1

Rs2 exp 2ikRs( )

3 ˆ η ⋅ ˆ R 2

=1Isotropic samples:

Paolo Fornasini

Univ. Trento

χ k( ) = 3 ˆ η ⋅ R 1

kR2 Im f k,π( ) e2iδ1 exp 2ikR( )[ ]

Single and multiple scattering

SS = Single scattering MS = Multiple scattering

Scattering paths

Paolo Fornasini

Univ. Trento

Multiple scattering series

Multiple Scattering ( ) ( ) ( ) ( ) ( )[ ]…++++= kkkkk 4320 1 χχχµµ

Single Scattering χ k( ) =µ − µ0µ0

( ) ( ) ( )[ ]kkk χµµ += 10

Paolo Fornasini

Univ. Trento

11 11.5 12Photon energy hν (keV)

µxµ0x

Multiple scattering paths

g4 g3 g2

gn = n-body correlation function

Multiple Scattering

µ k( ) = µ0 k( ) 1+ χ2 k( ) + χ3 k( ) + χ4 k( ) +…[ ]

Single Scattering χ k( ) =µ − µ0µ0

µ k( ) = µ0 k( ) 1+ χ k( )[ ]

Contribution from all n-order paths χn k( )

Paolo Fornasini

Univ. Trento

Multiple scattering contributions

µ k( ) = µ0 k( ) 1 + χ2 k( ) + χ3 k( ) +…χ n k( ) +…[ ]

Photo-electron path Set of coordinates

Functions of potential

χn k( ) = A k, r { }( ) sin kRp +φ k, r { }( )[ ] Neglecting

thermal disorder !

Paolo Fornasini

Univ. Trento

Intrinsic losses

Attenuation factor

µtot ω( ) = µ el ω( ) + µinel ω( )

χexp k( ) =µ −µ0

< χ th k( )

From experiment One-electron theory

S02 = Ψi

N−1 ΨfN−1 2

≈ 0.6 ÷ 0.9

Paolo Fornasini

Univ. Trento

χexp k( ) = S02 χ th k( )

Photo-electron mean-free-path

0 5 10 15 20

λ (Å

k (Å-1)

Core-hole lifetime τh → λh

Photo-electron lifetime τe ⇔ λe

exp − 2Rλ k( )

Paolo Fornasini

Univ. Trento

Attenuation factor

Disorder effects ?

EXAFS and inelastic effects

Atoms frozen in equilibrium positions !

χ k( ) =S02

shell∑ Im fs k,π( ) e2iδ1 e

−2Rs /λ (k )

Rs2 exp 2ikRs( )

Intrinsicinelastic effects

0 5 10 15k (Å-1)

χ (k)

Photo-electronmean-free-path

Paolo Fornasini

Univ. Trento

Local disorder

Distance Distribution of distances

Structural disorder - examples Thermal disorder

τvib ≈ 10-12 sτexafs ≈ 10-15 s

+•  Sites disorder

r•  Free-volume models

•  Distorted coordination shells

Paolo Fornasini

Univ. Trento

Thermal disorder in crystals

Separability of coordination shells ?

Paolo Fornasini

Univ. Trento

0 2 4 6 8r (Å)

Ge, 300 K

0 2 4 6 8r (Å)

Ge, 1200 K

Simulated distributions for c-Ge

Distributions of distances

Thermal + structural disorder

⇒ distribution of distances

Real distribution ρ(r)

Effective distributionP(r,λ)

χ k( ) =S02

shell∑ Im fs k,π( ) e2iδ1 ρs (r)

∫ e−2rs /λ (k )

rs2 e2ikrs dr

Paolo Fornasini

Univ. Trento

Real and effective distributions

0 2 4 6 8r (Å)

Ge, 300 K

0 2 4 6 8r (Å)

Ge, 1200 K

0 2 4 6 8r (Å)

Ge, 300 K

0 2 4 6 8r (Å)

Ge, 1200 K

Paolo Fornasini

Univ. Trento

ρ r( )

ρ r( )r2

P r,λ( ) =ρ r( )r2

e−2r /λ

(λ = 8Å)

EXAFS = short-range probe

The inversion problem Paolo

χ k( ) =S02

shell∑ Im fs k,π( ) e2iδ1 ρs (r)

∫ e−2rs /λ (k )

rs2 e2ikrs dr

Characteristic function

χ k( ) ⇒ ρ r( ) ?

Structural models and fitting procedure

ExperimentalEXAFS

Prametrisedstructural model Change

parameters

Compare with experiment

ρ (r)

Distributionof distances

kχ(k)∝ ρ(r) e−2r /λ

∫ e2ikrdr

Paolo Fornasini

Univ. Trento

Initial guessparameters

Calculate EXAFS

The simplest model: gaussian approximation Paolo Fornasini

Univ. Trento

P(r,λ) =1

σ 2πexp

r − r( )2

C2 = σ 2 = r − r( )2

Distribution width (EXAFS Debye-Waller factor)

Average distance

C1 = r eff = r real −2σ 2

Gaussian parametrization of EXAFS (one shell)

Coordination number

Debye-Waller

Average distance

k χ k( ) =S02 e−2C1 /λ

C12 f (k,π ) N exp −2k2σ 2[ ] sin 2kC1 + φ(k)[ ]

Inelasticterms

Totalphase-shift

Back-scatteringamplitude

•  Theory (interaction potentials + scattering theory) •  Experiment (reference samples)

Approx.: Single Scattering Plane waves

Paolo Fornasini

Univ. Trento

Including weak asymmetry Paolo Fornasini

Univ. Trento

Asymmetric distribution

C3 €

C2 = σ 2 = r − r( )2

C1 = r eff = r real −2σ 2

C3 = r − r( )3

Third cumulant Asymmetry parameter

Better for first shell

EXAFS including asymmetry (one shell)

Coordination number

Debye-Waller

Average distance and asymmetry

k χ k( ) =S02 e−2C1 /λ

C12 f (k,π ) N exp −2k2σ 2[ ] sin 2kC1 − 43 k

3C3 + φ(k)

Inelasticterms

Totalphase-shift

Paolo Fornasini

Univ. Trento

XAFS and other techniques Paolo

Photo-ionization

XPS - X-ray photo-electron spectroscopy

XAFS – X-ray absorption fine structure

Info on bound electronic states

Info on •  free electronic states •  local structure

XAFS = structural probe – Comparison with diffraction ?

Bragg diffraction .vs. EXAFS

Bragg diffraction EXAFS

Structural probe X-ray or neutron plane waves photo-electron spherical wave

Paolo Fornasini

Univ. Trento

•  long-range sensitivity •  atomic positions •  atomic thermal factors

•  short-range sensitivity •  inter-atomic distances •  relative displacements

EXAFS: a structural probe

Frequency

Inter-atomic distance

Amplitude

Coordinationnumber

Damping

Disorder

  Selectivity of atomic species   Insensitivity to long-range order

Paolo Fornasini

Univ. Trento

0 5 10 15k (Å-1)

R = 2 Å

R = 4 Å

0 5 10 15k (Å-1)

σ2 = 0.01 Å2

σ2 = 0.005 Å2

EXAFS applications

Non-crystalline materials

mono-atomic

many-atomic

•  Inorganic heterogeneos catalysts •  Metallo-proteins •  Impurities in semiconductors •  Luminescent atoms

Active sites embedded in a matrix

Local properties different from

average properties

•  Crystalline ternary random alloys •  Lattice dynamics studies •  Negative thermal expansion

Paolo Fornasini

Univ. Trento

EXAFS experiments

XAFS: experimental layout

• storage ring • beamlines

• monochromators • mirrors

•  sample holder • detectors

Sample conditioning: cryostat ovenreactor manipulators

Detection: transmissionfluorescenceelectron yield.......

sample holder

storage ring monochromator

mirror

detectors Optical apparatus Source Experim. apparatus

Alternative layouts • dispersive EXAFS • refl-EXAFS......

Paolo Fornasini

Univ. Trento

XAFS: experimental

♠ Monochromators and mirrors

X-ray crystal monochromators

Si (111) 6.2708Si (220) 3.84Si (311) 3.28Si (331) 2.5Si (511) 2.08 Ge (111) 6.5328Ge (220) 4.0004

Paolo Fornasini

Univ. Trento

- Forbidden ‘reflections’- Harmonics- Spurious reflections

Bragg law

2dhkl sinθ = nλ

Incidence angle ⇔ wavelength

Crystal reflectivity Paolo Fornasini

Univ. Trento

-1 0 3 4 1 2 Δθ (arc sec)

Darwinwidth

Rocking curve (from dynamical theory of diffraction)

Higher order reflections have narrower rocking curves.

Energy resolution

Darwin width (Intrinsic resolution) Total angle ΔΘ

Beam divergence

10 20 30E (keV)

Si (111)

ΔE (eV)

10 20 30

Si (220)

E (keV)

Darwin width

Core-levelwidth

=Δλλ

= ΔΘ cotgθB

Paolo Fornasini

Univ. Trento

Two-crystal monochromators

“Channel-cut”

  Mechanical simplicity Stability   Harmonics Spurious reflections

Independent crystals

  Detuning: harmonic reduction Possibility of focussing   Mechanical complexity Instability

Paolo Fornasini

Univ. Trento

Horizontal output beam

Crystals detuning Paolo Fornasini

Univ. Trento

(111) (333)

Silicon crystal (EB = 10 keV)

E-EB (eV)

3.5 arcsec

X-ray mirrors

θ c = 2δ ∝ λ ρ

harmonics rejection

grazing incidence θ ≈ 10-3 rad

n = 1−δ − iβ

δ ≈10−6 ÷ 10−5

for x - raysabsorption

Complex refractive index

Total external reflection : θ < θc

Beam collimation and

focalisation

Surface bending

Paolo Fornasini

Univ. Trento

XAFS: experimental

♠ Detection schemes

XAFS: direct transmission measurements

V ≈ 100 V/cm I ≈ 10-10 ÷ 10-8 A

Detectors: ionisation chambers

Paolo Fornasini

Univ. Trento

Sample: •  Powders or thin films •  Thickness ≈ 10 µm •  No holes or inhomogeneities

Ι0 Ι1

Direct transmission measurements Paolo Fornasini

Univ. Trento

Thick samples

Diluted samples

Surfaceinformation

Thin samples

Non-diluted samples

Homogeneous samples

Bulk information (not fom surface) from:

High accuracy attainable

Indirect detection methods Paolo Fornasini

Univ. Trento

•  X-ray fluorescence

Detection of decay products

•  Electrons

•  Optical luminescence XEOL-PLY =

FLY = FLuorescence Yield

AEY = Auger Electron Yield

PEY = Partial Eletron Yield

TEY = Total Electron Yield

X-ray Ecxited Optical Luminescence Photo Luminescence Yield

I f zn( ) dz = I0 (ω) exp −µs ω( )znsinθi

η fµa ω( ) dz

sinθiexp −

µs ω f( )znsinθ f

XAFS: fluorescence detection (FLY)

θi θf

Absorption Fluorescence Absorption

Paolo Fornasini

Univ. Trento

Fluorescence: total intensity

Sample of thickness zs θi = θf = 45°

Thin samples

1− exp A( ) ≈ 1−1− A = − A

I f ∝µa ω( )

Thick samples

1− exp A( ) ≈ 1

I f = I0 ω( )η fΩ4π

µa ω( )µs ω( ) + µs ω f( )

OK for diluted samples (< 1%)

Paolo Fornasini

Univ. Trento

I f = I0 ω( )η fΩ4π

µa ω( )µs ω( ) + µs ω f( )

1− exp A( ){ } A = − 2zs µs ω( ) + µs ω f( )[ ]

Fluorescence signal

Background signals

Elastic scattering Compton scattering Other fluorescences

Energy selective detectors Filters + Soller slits

Paolo Fornasini

Univ. Trento

XAFS: electron detection (a) Paolo Fornasini

Univ. Trento

Auger electrons: •  Fixed energy ⇒ atomic selectivity •  Intensity ∝ µx

⇒ XAFS signal

Photo-electrons: •  Energy varies with hν •  Intensity∝ µx

⇒ XAFS signal

Ι0 hν

Electron mean free path: •  adsorbates •  thin layers

Indirect processes and escape depth Paolo Fornasini

Univ. Trento

Photons penetration depth ~ 500 Å

Secondary electrons escape depth ~ 50 - 100 Å

Primary electrons mean free path ~ 5 - 10 Å

Adsorbate

XAFS: electron detection (b) Paolo Fornasini

Univ. Trento

AEY = Auger Electron Yield - narrow energy window - only direct Auger electrons - spurious structures from photoelectrons

PEY = Partial Eletron Yield - large energy window - Auger (direct + secondary) = XAFS signal - Photoel. (direct + secondary) = background

TEY = Total Electron Yield - all electrons collected

- Auger (direct + second.) = XAFS signal -  Photoel. (direct + second.) = background

-  XAFS from Auger and photoel.

XAFS of adsorbates

AEY - PEY - TEY

Bulk materials

XAFS: experimental

♠ Alternative layouts

Dispersive XAFS (a) Paolo Fornasini

Univ. Trento

Dispersive XAFS (b)

S.R. incoming white beam

Curved crystal poly-chromator

2d sinθ = λ

θ1 θ2 θ3

Position-sensitive detector

Paolo Fornasini

Univ. Trento

Dispersive XAFS (c)

 Critical in terms of temporal and spatial beam stability and sample presentation

 Only trasmission mode  X-ray beam not perfectly focussed through the sample  No reference measurements during acquisition

 No mechanical movements (no dead times)  Simultaneous acquisition of all data points  Acquisition time determined by acceptable statistics

Paolo Fornasini

Univ. Trento

OK for time-resolved measurements

NO accurate quantitative results

EXAFS: data analysis, examples

Analysis - Available software Paolo Fornasini

Univ. Trento

List of available software: XAFS Society web-site = http://xafs.org/Software

FEFF: ab initio MS calculations of EXAFS and XANES for clusters of atoms. The code yields scattering amplitudes and phases, as well as various other properties.

Athena: interactive graphical utility for processing EXAFS data. Artemis: interactive graphical utility for fitting EXAFS data using theoretical standards from FEFF and sophisticated data modelling.library.

IFEFFIT: interactive program for XAFS analysis.

GNXAS: EXAFS data analysis based on MS calculations and advanced fitting of raw experimental data. Main peculiarities: MS associated with 2, 3, and 4- atom configurations, multi-electron excitation, various model peaks for distribution functions.

EXAFS data analysis

♠ Extraction of EXAFS signal

Total absorption coefficient

source monochromator

sample detectors

Φ0 Φ

Paolo Fornasini

Univ. Trento

8800 9200 9600 10000E (eV)

CuCl (15 K)

edge Cu K

ln I0I

= lnΦ0

Φ+C' = µtotx+C'

Edge absorption coefficient

Ge, 10 K

Experimental signal

11 11.5 12

Photon energy (keV)

Extrapolationof pre-edge behaviour

Edge absorption coefficient

Paolo Fornasini

Univ. Trento

Photoelectron wavenumber Paolo Fornasini

Univ. Trento

Edgeenergy

k =2m2 hν −Es( )

Es E = hν, photon energy

Photoelectron wavenumber

Es = 1st maximum of 1st derivative Experimental convention

Atomic absorption coefficient Paolo Fornasini

Univ. Trento

EXAFS function

Isolated atom

Embedded atom

χ k( ) =µ −µ0

11 11.5 12Photon energy h ν (keV)

µxµ0x

Best-fitting polynomial spline Paolo Fornasini

Univ. Trento

0 500 1000 1500

Ge - 10 Kµx

E-Es (eV)

0 5 10 15 20

Ge - 10 Kµx

k (Å-1)

Polynomial spline - best fit

E space k space €

χ k( ) =µ −µ0

Fit optimization Paolo Fornasini

Univ. Trento

0 5 10 15 20

k (Å-1)

0 1 2 3 4 5

r (Å)

16Fit B

0 1 2 3 4 5r (Å)

0 5 10 15 20

µxFit B

k (Å-1)€

µ −µ0

Fourier transf.

EXAFS signal Paolo Fornasini

Univ. Trento

χ k( ) =µ −µ0

0 5 10 15 20

Ge - 10 K

k χ(k)

k (Å-1)

0.04Ge -10 K

χ (k)

EXAFS signals: examples

Amorphous Germanium

Crystalline Germanium

0 5 10 15k (Å-1)

0 5 10 15

k (Å-1)

T = 77 K T = 300 K T = 77 K

1 coord. shell Several coord. shells

Temperature effect

Paolo Fornasini

Univ. Trento

Diffraction .vs. EXAFS - (b)

0 4 8 12 16 20

0 2 4 6 8

c-SiO2

0 5 10 15 20 25

a-SiO2

G (1/A)0 5 10 15 20

0.4 a-Ge

k (1/A)

Diffraction EXAFS G k

Paolo Fornasini

Univ. Trento

Quantitative analysis of EXAFS Paolo Fornasini

Univ. Trento

0 5 10 15k (Å-1)

kχ(k)

χ(k) = Ai (k)sinΦi (k)i∑

Sum over: •  S.S. paths (coord. shells)

•  M.S. paths

Input for each path: •  backscattering amplitude •  phaseshifts •  inelastic terms Different analysis procedures

EXAFS data analysis

♠ Fourier transform

Data analysis - Fourier Transform k→r

5 10 15 20

k3 χ(k) W(k)

k (Å-1)

0 2 4 6r (Å)

Imaginary part

Modulus

Ge, 10 K

F(r) = χ k( ) kn W k( )kmin

∫ e2ikr dk

window weight

Peak's position and shape influenced by: - total phaseshifts - disorder - Fourier transform window

Paolo Fornasini

Univ. Trento

1.5 2 2.5 3 3.5

σ = 0.05 Å

σ = 0.1 Å

1.5 2 2.5 3 3.5r(A)

Fourier Transform and distribution Paolo Fornasini

Univ. Trento

F.T.: k=2.5-16 K3, square w.

EXAFS simulation (Ge phases and amplit.)

2 6 10 14

k)k (Å-1)

i Ni Ri (Å)1 8 2.482 6 2.863 12 4.054 24 4.755 8 4.966 6 5.73

0 1 2 3 4 5 6

r (Å)

Fe (T = 300K)

668 • Peak shift

• Superposition of shells

26 - Iron: bcc structure Paolo

0 1 2 3 4 5 6

r (Å)

Cu (T=4K )

i Ni Ri (Å)1 12 2.552 6 3.613 24 4.424 12 5.105 24 5.706 8 6.25

• Peak shift • Focussing effect

29 - Copper: fcc structure Paolo

0 5 10 15 20k (Å-1)

k2 χ(k

0 1 2 3 4 5 60

. of F

r (Å)

Fourier transforms EXAFS signals

29-Cu: temperature effects Paolo

i Ni Ri (Å)1 4 a(√3)/4 2.452 12 a/√2 4.003 12 a(√11)/4 4.694 6 a 5.665 12 a(√19)/4 6.166 24 a(√6)/2 6.93

0 1 2 3 4 5 6 7

r (Å)

Ge(10 K)

121212

32 - Germanium: diamond structure Paolo

a = 5.66 Å

Fourier transforms

EXAFS signals

32-Ge: crystalline and amorphous Paolo

0 2 4 6

r (Å)

10 K300 K

c - Ge

0 2 4 6r (Å)

10 K300 K

a - Ge

0 5 10 15 20k (Å-1)

10 K a-Ge

10 K c-Ge

0 5 10 15 20

k χ (

k (Å-1)

EXAFS data analysis

♠ First shell analysis

1st-shell Fourier back-transform Paolo Fornasini

Univ. Trento

0 1 2 3 4 5

Ge - 10 K

r (Å)

2 6 10 14 18

k (Å-1)

k χ(k)

χ ' k( ) = 2 π( ) F r( )W ' r( )rmin

rmax∫ e−2ikr dr

- No peak superposition- No MultipleScattering - F.T. artifacts

First-shell contribution

1st-shell distribution of distances Paolo Fornasini

Univ. Trento

Gaussian approximation

Asymmetric distribution

σ 2 = r − r( )2

C3 = r − r( )3

Better for first shell

EXAFS distance Paolo Fornasini

Univ. Trento

r = r b − r a

EXAFS, diffuse scattering

R = r b −

Bragg diffraction, dilatometry

Real and effective distributions Paolo Fornasini

Univ. Trento

Real distribution ρ(r)

Effective distributionP(r,λ)

ρs (r)e−2rs /λ (k )

r eff = r real −2σ 2

EXAFS for first shell

Coordination number

Debye-Waller

Average distance and asymmetry

k χ k( ) =S02 e−2C1 /λ

C12 f (k,π ) N exp −2k2σ 2[ ] sin 2kC1 − 43 k

3C3 + φ(k)

Inelasticterms

Totalphase-shift

Paolo Fornasini

Univ. Trento

Analysis - non-linear fitting method

Theoretical EXAFS

Experimental EXAFS

Theory

Paolo Fornasini

Univ. Trento

f k,π( ) , φ k( ), λ

e0, C1, C3

S02N, σ 2

Non-linear fit •  k-space •  r-space

•  Correlation •  Accuracy

•  abs. values •  rel. values

Check theory against standards

EXAFS data analysis

♠ 1st shell phase and amplitude analysis

Separate evaluation of phase and amplitude

Total phase (rad)

Amplitude

Paolo Fornasini

Univ. Trento

A k( ) =S02 e−2C1 /λ

C12 f (k,π ) N exp −2k2C2 +

24k 4C4 +…

Φ k( ) = 2kC1 −43k 3C3 + ...+φ k( )

From complex Fourier transform and back-transform

“Ratio method” - phases Paolo Fornasini

Univ. Trento

Φs −Φm = 2k C1s −C1

m( )− 43 k3 C3

s −C3m( )

Φs −Φm

2k= C1

s −C1m( )− 43 k

2 C3s −C3

If suitable model compound available …

Φs −Φm

ΔC3 = 0

ΔC3 > 0

s = sample m = model

“Ratio method” - amplitudes Paolo Fornasini

Univ. Trento

If suitable model compound available … s = sample m = model

Am≅ ln N

Nm − 2k2 σ s2 −σm

intercept Slope

Δσ 2

“Ratio method” - results Paolo Fornasini

Univ. Trento

ΔC1Δσ 2

Ratio of coordination numbers

Relative values :

→ Thermal expansion

Asymmetry   Absolute values ?   Physical meaning ?

“Ratio method” - OK when … Paolo

χ k( ) = A k( ) sinΦ k( )•  Only Single Scattering •  Only one distance •  Suitable reference model available

•  First coordination shell, one distance •  Same sample-model chemical environment

T or p-dep. Studies Amorphous .vs. crystalline samples

•  1st shell in bcc structure (2 distances) •  Superposed outer shells •  M.S. contributions

•  1st shell, different sample-model chemical environment

•  Separated outer shells, weak M.S. Depending on sought accuracy

EXAFS data analysis

♠ Outer shells analysis

Analysis - Outer shells back-transform r→k

2 6 10 14 18

k (Å-1)2 6 10 14 18

k (Å-1)

χ' k( ) = 2 π( ) F r( ) W' r( )rmin

∫ e−2ikr dr

0 1 2 3 4 5 6 r (Å)

Ge, 10 K

Paolo Fornasini

Univ. Trento

- Peak superposition- MultipleScattering - F.T. artifacts

Sometimes OK for D.W. factors

Analysis - non-linear fitting of outer shells

Theoretical EXAFS

Experimental EXAFS

Theory

Paolo Fornasini

Univ. Trento

fpath k,π( ) , φpath k( ), λ

e0, C1, C3

S02N, σ 2

Non-linear fit •  k-space •  r-space

•  Reduction of free param. •  Relations between param. •  Approximations

SS & MS path sorting

For every path !!

Δk Δr

N ind =2 Δk Δrπ

Maximum number of independent parameters

Analysis - Independent parameters Paolo Fornasini

Univ. Trento

Correlation ofparameters

EXAFS data analysis

♠ Interpretation of results

0 200 400 600

Cu - 1st shell

0-2 Å

XRDEXAFS

Thermal expansion Paolo

Bragg diffraction lattice expansion

EXAFS bond expansion

Complementarity EXAFS - XRD: Info on perpendicular vibrations

Mean Square Relative Displacements

Δu|| = 0

r ≈ R0 +Δu⊥

2R0σ 2 ≈ Δu||

2 MSRD||

MSRD⊥

Mean values (harmonic approximation)

r R0 €

Δu⊥

Paolo Fornasini

Univ. Trento

Bond distances Paolo Fornasini

Univ. Trento

r = r b − r a

EXAFS, diffuse scattering

“True” bond length “True” bond expansion

R = r b −

Bragg diffraction, dilatometry

“Apparent” bond length “Apparent” bond expansion

EXAFS Debye-Waller factor

MSDMean Square Displacements

DCFDisplacement

Correlation Function

σ 2 ≈ MSRD = Δu||2 = ˆ R ⋅ u b −

u a( )[ ] 2

= ˆ R ⋅ u b( )2

+ ˆ R ⋅ u a( )2− 2 ˆ R ⋅ u b( ) ˆ R ⋅ u a( )

Paolo Fornasini

Univ. Trento

Thermal factors from Bragg diffraction:

U||b = ˆ R ⋅ u b( )2

= σ ||b( )2

U||a = ˆ R ⋅ u a( )2

= σ ||a( )2

σ ||b

σ ||a

Debye-Waller factor – Debye model

0 100 200 300 400 500T (K)

10-2 Å2

Copper

1st shell

3rd shell4th shell2nd shell

Paolo Fornasini

Univ. Trento

Absolute values from fit to theoretical models

σ 2 =32ωD

3µω coth ω

ωD∫ 1−sin ωqDR( )ωqDR

Debye correlated model (OK for metals)

σ 2 •  increases with T •  depends on the shell

θD = 315 K θM = 313 K

θ4 = 321 K θ3 = 322 K θ2 = 283 K θ1 = 328 K

Correlation

0 100 200 300 400 500T (K)

10-2 Å2

Copper

1st shell

3rd shell4th shell2nd shell

Paolo Fornasini

Univ. Trento

Complementarity EXAFS - XRD: Info on vibrations correlation

Bragg diffraction absolute vibrations

EXAFS relative vibrations

(along the bond)

Debye-Waller factor – Einstein model

0 200 400

Germanium

Non-Bravais crystals

Δu||2 =

2µωE

coth ωE

θD = 354 K θM = 290 K

θ3 = 290 K θ2 = 299 K θ1 = 460 K

Debye Einstein

ν3 = 3.95 ν2 = 4.21 ν1 = 7.55 €

ν =ω / 2π(eV/Å2)

k3 = 2.18 k2 = 2.48 k1 = 8.15 €

k = µω2

Paolo Fornasini

Univ. Trento

Parallel and perpendicular MSRD

Δu⊥2

Δu||2

Paolo Fornasini

Univ. Trento

0 200 400 600

Cu 1st shell

0 200 400 600

Ge 1st shell

Δu⊥2

Δu||22.7 6 Perpendicular-parallel

anisotropy

First-shell distribution asymmetry

C3* T( ) ≈ − 2k3σ 0

k0z2 +10z+11− z( )2

Paolo Fornasini

Univ. Trento

0 200 400

C 3 (10-4

Cu - 1st shell

C3K3 = -1.37 eV/A3

The end

Basic bibliography Paolo Fornasini

Univ. Trento

•  G.S. Brown and S. Doniach: The principles of X-ray Absorption Spectroscopy, in Synchrotron Radiation research, ed. by E. Winick and S. Doniach, Plenum (New York, 1980) [A general introduction to X-Ray absorption]

• P.A. Lee, P.H. Citrin, P. Eisenberger, and B.M. Kincaid, Rev. Mod. Phys. 53, 769 (1981) [Review paper on EXAFS]

•  T.M. Hayes and J. B. Boyce, Solid State Physics 27, 173 (1982) [Review paper on EXAFS]

•  B.K. Teo: EXAFS, basic principles and data analysis, Springer (Berlin, 1986) [Introductory book on EXAFS]

•  D.C. Koningsberger and R. Prins eds.: X-ray Absorption: principles and application techniques of EXAFS, SEXAFS and XANES, J. Wiley (New York, 1988) [Introductory book on XAFS]

•  M. Benfatto, C.R. Natoli, and A. Filipponi, Phys. Rev. B 40, 926 (1989) [Paper on multiple scattering calculations]

•  J. Stöhr: NEXAFS spectroscopy, Springer (Berlin, 1996) [Book on XANES] •  J.J. Rehr and R.C. Albers, Rev. Mod. Phys. 72, 621 (2000) [Review paper on

EXAFS] • P. Fornasini, J. Phys.: Condens. Matter 13, 7859 (2001) [EXAFS and lattice

dynamics] • G. Bunker: Introduction to XAFS, Cambridge U.P. (2010) [Introductory book on XAFS]

XAFS Society home page: http://www.ixasportal.net/ixas/

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